6.5 : Polar Coordinate System
The polar coordinate system is formed by drawing a horizontal ray. The initial point of
the ray is called the pole. Take a look at these comparisons:
In the rectangular coordinate system we use (x, y) values to represent a location by mov-
ing xunits horizontally, and yunits vertically. In the polar coordinate system we use (r, θ),
where ris the distance to the point from the origin and 0 ≤θ < 2πto represent the angle
measure to that point.
Apolar equation is an equation in rand θand the solution to that equation is an ordered
pair (r, θ) that satisfies the equation. Take a look at a few examples on page 413 because
there’s a few nice pictures that demonstrate quite well.
Application: I really am not exaggerating when I say this information is incredibly useful
in engineering applications. Just in the last week I was using cardoids to help design optimal
placement of main speakers and subwoofers at my church here in town. Here’s an image of
the sound pattern created by subwoofers when aligned differently The one on the left is a
standard stereo setup with two channels to the left and right of the stage, whereas the right
image depicts a stacked setup with one facing one way, and another facing the opposite way..
These graphics are straight from the manufacturers website.
Transformations between Polar and Rectangular Coordinates
Given the point (r, θ) in the polar coordinate system, the transformation equations to change
from polar to rectangular coordinates are
x=rcos θ y =rsin θ
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